A Tauberian Theorem for Laplace Transforms with Pseudofunction Boundary Behavior

Complex Tauberian theorems for Laplace transforms with local pseudofunction boundary behavior

We provide several Tauberian theorems for Laplace transforms with local pseudofunction boundary behavior. Our results generalize and improve various known versions of the Ingham-Fatou-Riesz theorem and the Wiener-Ikehara theorem. Using local pseudofunction boundary behavior enables us to relax boundary requirements to a minimum. Furthermore, we allow possible null sets of boundary singularities and remove unnecessary uniformity conditions occurring in earlier works; to this end, we obtain a useful characterization of local pseudofunctions. Most of our results are proved under one-sided Tauberi…

Optimal Tauberian constant in Ingham's theorem for Laplace transforms

It is well known that there is an absolute constant $\mathfrak{C}>0$ such that if the Laplace transform $G(s)=\int_{0}^{\infty}ρ(x)e^{-s x}\:\mathrm{d}x$ of a bounded function $ρ$ has analytic continuation through every point of the segment $(-iλ,iλ)$ of the imaginary axis, then $$ \limsup_{x\to\infty} \left|\int_{0}^{x}ρ(u)\:\mathrm{d}u - G(0)\right|\leq \frac{ \mathfrak{C}}λ \: \limsup_{x\to\infty} |ρ(x)|. $$ The best known value of the constant $\mathfrak{C}$ was so far $\mathfrak{C}=2$. In this article we show that the inequality holds with $\mathfrak{C}=π/2$ and that this value is best po…

一个一般化的定量 Ingham-Karamata Tauber 定理

A general quantified Ingham-Karamata Tauberian theorem

本文提出一个一般化的定量 Ingham-Karamata Tauber 定理，采用灵活单边条件，改进 Stahn 定理并证明速率在几乎所有情形下最优。

Exponential decay of measures and Tauberian theorems

We study behavior of a measure on $[0,\infty)$ by considering its Laplace transform. If it is possible to extend the Laplace transform to a complex half-plane containing the imaginary axis, then the exponential decay of the tail of the measure occurs and under certain assumptions we show that the rate of the decay is given by the so called abscissa of convergence and extend the result of Nakagawa from [Nak05]. Under stronger assumptions we give behavior of density of the measure by considering its Laplace transform. In situations when there is no exponential decay we study occurrence of heavy …

An asymptotic analysis of the Fourier-Laplace transforms of certain oscillatory functions

We study the family of Fourier-Laplace transforms $$ F_{α,β}(z)= \operatorname*{F.p.} \int_{0}^{\infty} t^β\exp(\mathrm{i} t^α-\mathrm{i} z t)\:\mathrm{d} t, \quad \operatorname*{Im} z<0, $$ for $α>1$ and $β\in\mathbb{C}$, where Hadamard finite part is used to regularize the integral when $\operatorname*{Re} β\leq -1$. We prove that each $F_{α,β}$ has analytic continuation to the whole complex plane and determine its asymptotics along any line through the origin. We also apply our ideas to show that some of these functions provide concrete extremal examples for the Wiener-Ikehara theorem and a…

Multidimensional Tauberian theorems for vector-valued distributions

We prove several Tauberian theorems for regularizing transforms of vector-valued distributions. The regularizing transform of $f$ is given by the integral transform $M^{f}_φ(x,y)=(f\astφ_{y})(x),$ $(x,y)\in\mathbb{R}^{n}\times\mathbb{R}_{+}$, with kernel $φ_{y}(t)=y^{-n}φ(t/y)$. We apply our results to the analysis of asymptotic stability for a class of Cauchy problems, Tauberian theorems for the Laplace transform, the comparison of quasiasymptotics in distribution spaces, and we give a necessary and sufficient condition for the existence of the trace of a distribution on $\left\{x_0\right\}\t…